 This study investigates the impact of curvature on reaction diffusion waves on curved two-dimensional surfaces using the Fitz-Uniguma model. It shows that the stability of propagating wave segments depends on the curvature and can shrink or expand depending on whether they are smaller or larger than a critical nucleus modified by the curvature acting like an effective space-dependent local spatial coupling. A negative gradient of Gaussian curvature allows for stable propagation of localized wave segments remaining unchanged in size and shape or oscillating periodically in size. This article was authored by Fredo Agneer, Eckhart Scholl, and Marcus A. Darlem.